xrge0iifiso - Mathbox for Thierry Arnoux

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Theorem xrge0iifiso 34270

Description: The defined bijection from the closed unit interval onto the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)

**Hypothesis**
Ref	Expression
xrge0iifhmeo.1	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))

**Assertion**
Ref	Expression
xrge0iifiso	⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Distinct variable group: 𝑥,𝐹

Proof of Theorem xrge0iifiso Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 13457	. . 3 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 13166	. . 3 ⊢ < Or ℝ^*
3		soss 5590	. . 3 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. 2 ⊢ < Or (0[,]1)
5		iccssxr 13457	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
6		cnvso 6290	. . . . 5 ⊢ ( < Or ℝ^* ↔ ^◡ < Or ℝ^*)
7	2, 6	mpbi 233	. . . 4 ⊢ ^◡ < Or ℝ^*
8		sopo 5589	. . . 4 ⊢ (^◡ < Or ℝ^* → ^◡ < Po ℝ^*)
9	7, 8	ax-mp 5	. . 3 ⊢ ^◡ < Po ℝ^*
10		poss 5572	. . 3 ⊢ ((0[,]+∞) ⊆ ℝ^* → (^◡ < Po ℝ^* → ^◡ < Po (0[,]+∞)))
11	5, 9, 10	mp2 9	. 2 ⊢ ^◡ < Po (0[,]+∞)
12		xrge0iifhmeo.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
13	12	xrge0iifcnv 34268	. . . 4 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 = +∞, 0, (exp‘-𝑧))))
14	13	simpli 488	. . 3 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
15		f1ofo 6829	. . 3 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) → 𝐹:(0[,]1)–onto→(0[,]+∞))
16	14, 15	ax-mp 5	. 2 ⊢ 𝐹:(0[,]1)–onto→(0[,]+∞)
17		0xr 11256	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
18		1xr 11268	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
19		0le1 11737	. . . . . . . 8 ⊢ 0 ≤ 1
20		snunioc 13507	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
21	17, 18, 19, 20	mp3an 1487	. . . . . . 7 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
22	21	eleq2i 2861	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ 𝑤 ∈ (0[,]1))
23		elun 4113	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
24	22, 23	bitr3i 280	. . . . 5 ⊢ (𝑤 ∈ (0[,]1) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
25		velsn 4608	. . . . . . 7 ⊢ (𝑤 ∈ {0} ↔ 𝑤 = 0)
26		elunitrn 13494	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ∈ ℝ)
27	26	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ ℝ)
28		simpr 489	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 0 < 𝑧)
29		elicc01 13493	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
30	29	simp3bi 1163	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ≤ 1)
31	30	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ≤ 1)
32		1re 11208	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
33		elioc2 13436	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1)))
34	17, 32, 33	mp2an 704	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1))
35	27, 28, 31, 34	syl3anbrc 1360	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ (0(,]1))
36		pnfxr 11263	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
37		0le0 12342	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 0
38		ltpnf 13145	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℝ → 1 < +∞)
39	32, 38	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 1 < +∞
40		iocssioo 13466	. . . . . . . . . . . . . . 15 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
41	17, 36, 37, 39, 40	mp4an 705	. . . . . . . . . . . . . 14 ⊢ (0(,]1) ⊆ (0(,)+∞)
42		ioorp 13452	. . . . . . . . . . . . . 14 ⊢ (0(,)+∞) = ℝ⁺
43	41, 42	sseqtri 3991	. . . . . . . . . . . . 13 ⊢ (0(,]1) ⊆ ℝ⁺
44	43	sseli 3939	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0(,]1) → 𝑧 ∈ ℝ⁺)
45		relogcl 26706	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → (log‘𝑧) ∈ ℝ)
46	45	renegcld 11641	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) ∈ ℝ)
47		ltpnf 13145	. . . . . . . . . . . . . 14 ⊢ (-(log‘𝑧) ∈ ℝ → -(log‘𝑧) < +∞)
48	46, 47	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) < +∞)
49		brcnvg 5866	. . . . . . . . . . . . . 14 ⊢ ((+∞ ∈ ℝ^* ∧ -(log‘𝑧) ∈ ℝ) → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
50	36, 46, 49	sylancr 598	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
51	48, 50	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → +∞^◡ < -(log‘𝑧))
52	44, 51	syl 18	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < -(log‘𝑧))
53	12	xrge0iifcv 34269	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → (𝐹‘𝑧) = -(log‘𝑧))
54	52, 53	breqtrrd 5141	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < (𝐹‘𝑧))
55	35, 54	syl 18	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → +∞^◡ < (𝐹‘𝑧))
56	55	ex 417	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,]1) → (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧)))
57		breq1 5114	. . . . . . . . 9 ⊢ (𝑤 = 0 → (𝑤 < 𝑧 ↔ 0 < 𝑧))
58		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = (𝐹‘0))
59		0elunit 13496	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
60		iftrue 4496	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
61		pnfex 11262	. . . . . . . . . . . . 13 ⊢ +∞ ∈ V
62	60, 12, 61	fvmpt 6990	. . . . . . . . . . . 12 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
63	59, 62	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹‘0) = +∞
64	58, 63	eqtrdi 2820	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = +∞)
65	64	breq1d 5121	. . . . . . . . 9 ⊢ (𝑤 = 0 → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ +∞^◡ < (𝐹‘𝑧)))
66	57, 65	imbi12d 347	. . . . . . . 8 ⊢ (𝑤 = 0 → ((𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)) ↔ (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧))))
67	56, 66	imbitrrid 249	. . . . . . 7 ⊢ (𝑤 = 0 → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
68	25, 67	sylbi 220	. . . . . 6 ⊢ (𝑤 ∈ {0} → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
69		simpll 778	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ (0(,]1))
70	26	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ ℝ)
71		0re 11210	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
72	71	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 ∈ ℝ)
73	43	sseli 3939	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ⁺)
74	73	rpred 13060	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ)
75	74	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ ℝ)
76		elioc2 13436	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1)))
77	17, 32, 76	mp2an 704	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1))
78	77	simp2bi 1162	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 0 < 𝑤)
79	78	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑤)
80		simpr 489	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 < 𝑧)
81	72, 75, 70, 79, 80	lttrd 11371	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑧)
82	30	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ≤ 1)
83	70, 81, 82, 34	syl3anbrc 1360	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ (0(,]1))
84	69, 83	jca 520	. . . . . . . 8 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)))
85	73	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑤 ∈ ℝ⁺)
86	85	relogcld 26754	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑤) ∈ ℝ)
87	44	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑧 ∈ ℝ⁺)
88	87	relogcld 26754	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑧) ∈ ℝ)
89	86, 88	ltnegd 11792	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((log‘𝑤) < (log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
90		logltb 26731	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
91	73, 44, 90	syl2an 607	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
92		negex 11455	. . . . . . . . . . . . 13 ⊢ -(log‘𝑤) ∈ V
93		negex 11455	. . . . . . . . . . . . 13 ⊢ -(log‘𝑧) ∈ V
94	92, 93	brcnv 5869	. . . . . . . . . . . 12 ⊢ (-(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤))
95	94	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (-(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
96	89, 91, 95	3bitr4d 314	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
97	96	biimpd 232	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → -(log‘𝑤)^◡ < -(log‘𝑧)))
98	12	xrge0iifcv 34269	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0(,]1) → (𝐹‘𝑤) = -(log‘𝑤))
99	98, 53	breqan12d 5127	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
100	97, 99	sylibrd 262	. . . . . . . 8 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
101	84, 80, 100	sylc 66	. . . . . . 7 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
102	101	exp31 424	. . . . . 6 ⊢ (𝑤 ∈ (0(,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
103	68, 102	jaoi 870	. . . . 5 ⊢ ((𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
104	24, 103	sylbi 220	. . . 4 ⊢ (𝑤 ∈ (0[,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
105	104	imp 411	. . 3 ⊢ ((𝑤 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
106	105	rgen2 3211	. 2 ⊢ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
107		soisoi 7327	. 2 ⊢ ((( < Or (0[,]1) ∧ ^◡ < Po (0[,]+∞)) ∧ (𝐹:(0[,]1)–onto→(0[,]+∞) ∧ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))) → 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞)))
108	4, 11, 16, 106, 107	mp4an 705	1 ⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∪ cun 3909 ⊆ wss 3911 ifcif 4490 {csn 4592 class class class wbr 5111 ↦ cmpt 5194 Po wpo 5568 Or wor 5569 ^◡ccnv 5661 –onto→wfo 6535 –1-1-onto→wf1o 6536 ‘cfv 6537 Isom wiso 6538 (class class class)co 7411 ℝcr 11099 0cc0 11100 1c1 11101 +∞cpnf 11240 ℝ^*cxr 11242 < clt 11243 ≤ cle 11244 -cneg 11442 ℝ⁺crp 13016 (,)cioo 13372 (,]cioc 13373 [,]cicc 13375 expce 16115 logclog 26685

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3375 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ioc 13377 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15104 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-limsup 15522 df-clim 15539 df-rlim 15540 df-sum 15738 df-ef 16121 df-sin 16123 df-cos 16124 df-pi 16126 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-lp 23262 df-perf 23263 df-cn 23353 df-cnp 23354 df-haus 23441 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-limc 25994 df-dv 25995 df-log 26687

This theorem is referenced by: xrge0iifhmeo 34271

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